Interpreting Distance-Time Graphs

CONCEPT DEVELOPMENT

Mathematics Assessment Project

CLASSROOM CHALLENGES

A Formative Assessment Lesson

Interpreting Distance-Time Graphs

Mathematics Assessment Resource Service University of Nottingham & UC Berkeley Beta Version

For more details, visit: ? 2012 MARS, Shell Center, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at - all other rights reserved

Interpreting Distance?Time Graphs

MATHEMATICAL GOALS

This lesson unit is intended to help you assess how well students are able to interpret distance?time graphs and, in particular, to help you identify students who:

? Interpret distance?time graphs as if they are pictures of situations rather than abstract representations of them.

? Have difficulty relating speeds to slopes of these graphs.

COMMON CORE STATE STANDARDS

This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards for Mathematics:

8.F Construct a function to model a linear relationship between two quantities. Describe qualitatively the functional relationship between two quantities by analyzing a graph

This lesson also relates to the following Standards for Mathematical Practice in the Common Core State Standards for Mathematics:

2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others.

INTRODUCTION

The lesson unit is structured in the following way:

? Before the lesson, students work on a task designed to reveal their current understandings and difficulties. You review their work and create questions for students to answer in order to improve their solutions.

? A whole-class introduction provides students with guidance on how to work through the first task. Students then work in small groups on a collaborative discussion task, matching verbal interpretations with graphs. As they do this, they translate between words and graphical features, and begin to link the representations.

? This is followed by a whole-class discussion about applying realistic data to a graph. ? Students next work in small groups, matching tables of data to the existing matched pairs of

cards. They then explain their reasoning to another group of students. ? In a final whole-class discussion, students draw their own graphs from verbal interpretations. ? Finally, students return to their original task and try to improve their individual responses.

MATERIALS REQUIRED

? Each student will need two copies of the assessment task Journey to the Bus Stop, a miniwhiteboard, a pen, and an eraser.

? Each small group of students will need copies of Card Set A: Distance?Time Graphs, Card Set B: Interpretations, Card Set C: Tables of Data, a large sheet of paper, and a glue stick for making posters. The cards should be cut up beforehand.

? You will also need a supply of graph paper to give to students who request it. There are some projector resources to support your teaching.

TIME NEEDED

15 minutes before the lesson, a 90-minute lesson (or two 45-minute lessons), and 10 minutes in a following lesson (or homework). Timings are approximate and will depend on the needs of the class.

Teacher guide

Interpreting Distance-Time Graphs

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BEFORE THE LESSON

Assessment task: Journey to the Bus Stop (15 minutes)

Set this task, in class or for homework, a few days

Interpreting Distance?Time Graphs

Student Materials

Alpha 2 version 21 Oct 2010

before the formative assessment lesson. This will

Journey to the Bus Stop

give you an opportunity to assess the work, and to

Every morning Tom walks along a straight road from his home to a bus stop, a distance of 160 meters. The graph shows his journey on one particular day.

find out the kinds of difficulties students have with

it. You will then be able to target your help more

effectively in the follow-up lesson.

Give each student a copy of Journey to the Bus Stop.

Briefly introduce the task and help the class to understand the problem and its context.

Read through the task and try to answer it as carefully as you can.

1. Describe what may have happened. You should include details like how fast he walked.

It is important that, as far as possible, students are allowed to answer the questions without your assistance.

Students should not worry too much if they cannot

2. Are all sections of the graph realistic? Fully explain your answer.

understand or do everything because in the next

lesson they will engage in a similar task that should

help them. Explain to students that by the end of the

next lesson, they should expect to answer questions

? 2010 Shell Center/MARS University of Nottingham UK

S-1

such as these confidently. This is their goal.

Assessing students' responses Collect students' responses to the task. Make some notes on what their work reveals about their current levels of understanding and their different problem solving approaches.

We suggest that you do not score students' work. The research shows that this will be counterproductive, as it will encourage students to compare their scores and will distract their attention from what they can do to improve their mathematics.

Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this unit.

We suggest that you write a list of your own questions, based on your students' work, using the ideas that follow. You may choose to write questions on each student's work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the end of the lesson.

Teacher guide

Interpreting Distance-Time Graphs

T-2

Common issues:

Suggested questions and prompts:

Student interprets the graph as a picture

For example: The student assumes that as the graph goes up and down, Tom's path is going up and down.

Or: The student assumes that a straight line on a graph means that the motion is along a straight path.

Or: The student thinks the negative slope means Tom has taken a detour.

? If a person walked in a circle around their home, what would the graph look like?

? If a person walked at a steady speed up and down a hill, directly away from home, what would the graph look like?

? In each section of his journey, is Tom's speed steady or is it changing? How do you know?

? How can you figure out Tom's speed in each section of the journey?

Student interprets graph as speed?time

The student has interpreted a positive slope as speeding up and a negative slope as slowing down.

? If a person walked for a mile at a steady speed, away from home, then turned round and walked back home at the same steady speed, what would the graph look like?

? How does the distance change during the second section of Tom's journey? What does this mean?

? How does the distance change during the last section of Tom's journey? What does this mean?

? How can you tell if Tom is traveling away from or towards home?

Student fails to mention distance or time

For example: The student has not mentioned how far away from home Tom has traveled at the end of each section.

Or: The student has not mentioned the time for each section of the journey.

Student fails to calculate and represent speed

For example: The student has not worked out the speed of some/all sections of the journey.

Or: The student has written the speed for a section as the distance covered in the time taken, such as "20 meters in 10 seconds."

Student misinterprets the scale

For example: When working out the distance the student has incorrectly interpreted the vertical scale as going up in 10s rather than 20s.

Student adds little explanation as to why the graph is or is not realistic

? Can you provide more information about how far Tom has traveled during different sections of his journey?

? Can you provide more information about how much time Tom takes during different sections of his journey?

? Can you provide information about Tom's speed for all sections of his journey?

? Can you write his speed as meters per second?

? What is the scale on the vertical axis?

? What is the total distance Tom covers? Is this realistic for the time taken? Why?/Why not?

? Is Tom's fastest speed realistic? Is Tom's slowest speed realistic? Why?/Why not?

Teacher guide

Interpreting Distance-Time Graphs

T-3

SUGGESTED LESSON OUTLINE

If you have a short lesson or you find the lesson is progressing at a slower pace than anticipated, we suggest you break the lesson after the first sharing of posters and continue it at a later time.

Whole-class introduction: interpreting and sketching graphs (10 minutes) Throughout this activity, encourage students to articulate their reasoning, justify their choices mathematically, and question the choices put forward by others. This introduction will provide students with a model of how they should work with their partners in the first small-group activity.

Show the class the projector resource Matching a Graph to a Story:

Matching a Graph to a Story

A. Tom took his dog for a walk to the park. He set off slowly and then increased his pace. At the park Tom turned around and walked slowly back home.

B. Tom rode his bike east from his home up a steep hill. After a while the slope eased off. At the top he raced down the other side.

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C. Tom went for a jog. At the end of his road he bumped into a friend and his pace slowed. When Tom left his friend he walked quickly back home.

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Projector Resources

Interpreting Distance-Time Graphs

P-1

Ask students to match the correct story to the graph. They are to write down at least two reasons to

support their decision.

After two or three minutes ask students who selected option A to raise their hands. Ask one or two to justify their choice. You may wish to use some of the questions on the sheet Suggested questions and prompts to encourage students to justify their choices and others to challenge their reasoning.

Repeat this with options B and C. Even if explanations are incorrect or only partially correct, write them next to the appropriate section of the graph. Encourage students to challenge these interpretations.

Slide P-2 of the projector resource allows you to write three different student explanations on the board at the same time.

Teacher guide

Interpreting Distance-Time Graphs

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